$\alpha_p(S,T)$ is the $p$-adic density of the representation of $T$ by $S$ ($p$ over the rational primes),

and
$$
n\leq r,\qquad n\leq m-r,\qquad 2(n+1) < m.
$$

In my case, $n=1$ thus $T=t$ is just a (non zero) integer number, and $S$ has signature $(m-1,1)$ (or $(1,m-1)$). Hence, the theorem holds for
$$
m>4.
$$
However, in the fourth page, he added that the formula also holds for $m=4$.

I known that when $m=3$ ($S$ is a ternary quadratic form), the formula doesn't holds in general and strange things happen (see BOROVOI ).

My question is: Does the formula works for the following particular case?:

2 Answers
2

Borovoi does not claim what you seem to think. Siegel's formalism discusses, for positive forms, the number of representations of an integer by an entire genus of positive ternary forms, each form weighted according to the number of its integral automorphs. For indefinite forms, the automorph groups are infinite and one must discuss the number of essentially inequivalent representations of an integer by a form, that is representations that cannot be taken to each other by automorphs.

That being said, most indefinite ternary forms are in a genus with only one integer equivalence class of forms. In such cases, Siegel's answers for a genus agree with those for the single form.

Borovoi simply repeats one of the more famous examples of a genus of more than one class. In the first edition (1988) of SPLAG by Conway and Sloane, this is on pages 404-405. The two classes are represented by
$$ x^2 - 2 y^2 + 64 z^2 $$ and
$$ -9 x^2 + 2 x y + 7 y^2 + 2 z^2. $$
In at least three variables, with indefinite forms, spinor genera coincide with equivalence classes. That is, every indefinite ternary form is spinor regular. There is, as there is here, a finite set of squareclasses of "spinor exceptional integers" that may fail to be represented by a form, even though another form in the genus represents them. Borovoi does not mention it in this manner (he takes only my $ \; s=1$), but here we go:

Let us define a (positive) integer $s$ such that all prime factors $q$ of $s$ satisfy
$$ q \equiv \pm 1 \pmod 8, $$
with the consequence that also $ s \equiv \pm 1 \pmod 8. $
Then, in notation going back to Jones and Pall (1939), Borovoi's form does not integrally represent $s^2,$ as in
$$ -9 x^2 + 2 x y + 7 y^2 + 2 z^2 \neq s^2.$$
Borovoi gives all necessary steps. I should say already that every example I know of spinor exceptional integers can be proved, after the fact, by very simple factoring arguments.

Will, it's a nice answer! Thank you very much. However, I want to check my conclusion from your answer in the next comment.
–
emiliocbaFeb 4 '12 at 0:13

Are you saying that if the ternary form $S=diag(A,-a)$ (with $A$ pos def and $a\in\mathbb N$) is in a genus with only one integer equivalence class, then Siegel's formula holds?
–
emiliocbaFeb 4 '12 at 0:16

@emiliocba, yes. What is your actual background as relates to this material?
–
Will JagyFeb 4 '12 at 1:46

Thank you again. I want to known if these particular case holds generalizing the fact that it holds for $S=diag(1,1,-1)$ and $t<0$.
–
emiliocbaFeb 4 '12 at 17:34

This next bit is unexpected, at least in terms of finding an explicit formula. We get a cheap trick out of $$ -9 x^2 + 2 x y + 7 y^2 = (9x + 7 y)(-x+y). $$ If we take $y=x+1$ the result of the binary is $16 x + 7.$ Add on $2 z^2$ with $z=1$ we actually get $16 x + 9.$

Though you should build a bark of dead men's bones,
And rear a phantom gibbet for a mast,
Stitch creeds together for a sail, with groans
To fill out, blood-stained and aghast;
Although your rudder be a dragon's tail
Long sever'd, yet still hard with agony
Your cordage large uprootings from the skull
Of bald Medusa, certes you would fail
To find the Melancholy - whether she
Dreameth in any isle of Lethe dull.

Why do you quote a melancholic Ode, / And format it as though 'twere pseudocode?
–
Noam D. ElkiesFeb 5 '12 at 3:39

@Noam, $$ $$ A writer claimed this verse composed, /// Not by Keats, as all supposed, /// But by his landlord, Charles Brown, /// Who wrote his tenant's stanzas down. /// The British Library says not so, /// Revealed by Stephen Hebron. Lo, /// I tried the Latex verse environ, /// But MO balked, and said, no Byron, /// Nor Coleridge shall our site pollute, /// Lest I suggest I did compute /// The lines I liked. So I gave in. /// As oft I've noticed, I can't win.
–
Will JagyFeb 5 '12 at 6:00

$$ \begin{array}{l} \text{To format text like poetry,}\phantom{XXXXXXXXXXXXXXXXXXXXXXXXXXXX}\cr \text{Use not quadruple-space, but <pre>.} \end{array} $$ (And in comments, do something like "begin{array}{l}\text{There once was a man in Peru,}\cr\text{Whose limericks stopped at line two.}end{array}" between double dollar marks, and with backslashes before the "begin" and "end".)
–
Noam D. ElkiesFeb 5 '12 at 22:43

@Noam, Excellent. I'll send you the book review and book pages I meant, 2 jpegs and a pdf.
–
Will JagyFeb 5 '12 at 23:13